← Back to feed

A preprint posted to arXiv on August 27 claims to have resolved the Spherical Hadwiger Conjecture, an open problem in integral geometry since approximately 1974. The authors are Wang Suijie and co-researchers at Hunan University. OpenAI Codex is listed as a contributor to proof development.

The paper’s disclosure is explicit: Codex was used “to assist with developing proof details, identifying gaps and points requiring clarification, organizing and typesetting the manuscript, and editing the English.” The authors reviewed and verified all AI-assisted content and take full responsibility for the result.

What Was Proved

The Spherical Hadwiger Theorem, as stated in the paper, classifies every continuous SO(n+1)-invariant valuation on the space of all closed spherical convex sets in the n-sphere. The result says that any such valuation can be written uniquely as a linear combination of the spherical intrinsic volumes V₀ through Vₙ. The proof is inductive and relies on a unique extension to non-proper sets, a continuous alternating cocycle on oriented spherical simplices, and a signed coning transform.

The companion result, via the cone-sphere correspondence, classifies continuous SO(d)-invariant conic valuations on all closed convex cones in ℝᵈ for d ≥ 2.

The preprint is 28 pages. Community verification is ongoing.

The Context

The week the Hadwiger preprint landed, Anthropic published Claude Fable’s formalization of Fermat’s Last Theorem in Lean — 13 million lines of code, 29,500 intermediate proofs, 11 days of autonomous operation. The two results are structurally distinct: Claude’s FLT work was formalization of an existing proof (machine-checking every step of Wiles’ 1995 argument from first axioms), while the Hadwiger result is a new proof of a previously unsolved problem, with AI contributing to the proof’s actual content.

That distinction matters. Formalization confirms what mathematicians already believe to be true. A new proof — even one with AI-developed details — adds to the stock of mathematical knowledge. If the Hadwiger result is verified, it is the latter.

Both papers cite adherence to the Leiden Declaration, a set of principles for responsible AI use in academic research published in 2025.

The Pattern

The Hadwiger paper is not the first AI-assisted resolution of a long-open problem. GPT-5.6 Sol completed a formal proof of the Erdős Unit Distance Conjecture in Lean earlier this year. GPT-5.4 Pro closed an Erdős problem from 1968 in April. GPT-5.6 Sol Ultra produced a proof of the Cycle Double Cover Conjecture in July.

What distinguishes the Hadwiger result is the proof mechanism. Earlier results were either formalizations of known arguments or single-model outputs where the AI’s contribution was verifiable after the fact. Here, Codex was embedded in the proof-creation workflow itself — developing details that, by the authors’ own description, would previously have required a research-level mathematician.

The Spherical Hadwiger Conjecture is not a household-name problem. It sits in a specialized corner of metric geometry. That is, in some ways, the point: the cases where AI is now sufficient to close long-open problems are no longer limited to problems with high economic or cultural salience. The technology is working its way through the backlog.